The Australia international, 25, is out of contract in the summer, but wants to follow in Argentinian striker Luciano Becchio's footsteps by signing a new long-term contract.
Kilkenny told the Yorkshire Evening Post: "I really want to stay here and feel honoured to play for the club.
"I want to let that be known. I've enjoyed my time here and feel that I've played regularly over the past three years.
"I've made nearly 150 appearances for the club now and that's really a very good record over three seasons. Hopefully, the club can sort something out."
Meanwhile, Kilkenny could miss out on Leeds' FA Cup clash with former club Arsenal in January if he is named in Australia's 23-man squad to be announced on December 28.